Question

B14 Из пункта А в пункт В одновременно выехали два автомобиля. Первый проехал с постоянной скоростью весь путь. второй проехал первую половину пути со скоростью, меньшей скорости первого на 13 км/ч, а вторую половину пути- со скоростью 78 км/ч, в результате чего прибыл в пункт В одновременно с первым автомобилем. Найдите скорость первого автомобиля, если известно, что она больше 48 км/ч.

Answer 1

65 км/ч
Проверка: 65-13 = 52 км/ч
(52 + 78)/2 = 65 км/ч

Answer 2

Х - скорость первого автомобиля. 
L - расстояние между пунктами. 
(Х+22) - скорость 2 автом. на втором участке. 
Тогда с учетом условия: 
L/Х - время движения 1 автомобиля 
0,5L/33+0,5L/(Х+22) - время движения 2 астом. 
По условию они равны. 
L/Х =0,5L/33+0,5L/(Х+22) 
1/Х=1/66+1/(2Х+44). Умножаем обе части на 66*Х*(Х+22) и избавляемся от знаменателя. Имеем: 
66*(Х+22)=Х*(Х+22)+33*Х. 
Раскрываем скобки и переносим все в правую часть. 
Х^2+22Х+33Х-66Х-1452=0 (Х^2 - это Х в квадрате) 
Х^2-11Х-1452=0. Решаем квадратное уравнение 
Х1= 11/2+кор. квадр из [(11/2)^2+1452]=44 (км/час.) 
Х2=11/2-кор. квадр из [(11/2)^2+1452]<0 - не имеет смысла 
Ответ: Х=44 км/час.

Answer 3

Problem Analysis

We are given that two cars simultaneously traveled from point A to point B. The first car traveled the entire distance at a constant speed. The second car traveled the first half of the distance at a speed 13 km/h slower than the first car, and the second half of the distance at a speed of 78 km/h. The second car arrived at point B at the same time as the first car. We need to find the speed of the first car, knowing that it is greater than 48 km/h.

Solution

Let's assume the distance from point A to point B is d km.

The first car traveled the entire distance at a constant speed, which we'll call v km/h. Therefore, the time taken by the first car to travel from point A to point B is given by:

Time taken by first car = d/v (Equation 1)

The second car traveled the first half of the distance at a speed 13 km/h slower than the first car, which means its speed was v - 13 km/h. The time taken by the second car to travel the first half of the distance is given by:

Time taken by second car for first half = (d/2) / (v - 13) (Equation 2)

The second car traveled the second half of the distance at a speed of 78 km/h. The time taken by the second car to travel the second half of the distance is given by:

Time taken by second car for second half = (d/2) / 78 (Equation 3)

Since the second car arrived at point B at the same time as the first car, the total time taken by the second car is equal to the time taken by the first car. Therefore, we can equate the sum of the times taken by the second car for the first and second halves to the time taken by the first car:

(d/2) / (v - 13) + (d/2) / 78 = d/v (Equation 4)

Now, we can solve Equation 4 to find the value of v.

Calculation

Let's solve Equation 4 to find the value of v:

(d/2) / (v - 13) + (d/2) / 78 = d/v

Multiplying both sides of the equation by 2v(v - 13) and simplifying, we get:

v(v - 13) + (v - 13)(78) = 2d

Expanding and rearranging the equation, we get:

v^2 - 13v + 78v - 1014 = 2d

v^2 + 65v - 1014 - 2d = 0

Now, we substitute the value of d as d = 2v into the equation:

v^2 + 65v - 1014 - 2(2v) = 0

v^2 + 65v - 1014 - 4v = 0

v^2 + 61v - 1014 = 0

Now, we can solve this quadratic equation to find the value of v.

Using the quadratic formula, we have:

v = (-b ± √(b^2 - 4ac)) / (2a)

where a = 1, b = 61, and c = -1014.

Substituting the values, we get:

v = (-61 ± √(61^2 - 4(1)(-1014))) / (2(1))

Simplifying further, we get:

v = (-61 ± √(3721 + 4056)) / 2

v = (-61 ± √(7777)) / 2

Since the speed of the first car is greater than 48 km/h, we can discard the negative value of v.

Therefore, the speed of the first car is:

v = (-61 + √(7777)) / 2 (approximately 43.47 km/h)

Answer

The speed of the first car is approximately 43.47 km/h.